Explanation Given data: F ( x , y ) = ( 4 x 3 y 2 − 2 x y 3 ) i + ( 2 x 4 y − 3 x 2 y 2 + 4 y 3 ) j (1) C : r ( t ) = ( t + sin π t ) i + ( 2 t + cos π t ) j , 0 ≤ t ≤ 1 (2) Formula used: Consider the standard equation of curl F for F = P i + Q j + R k c u r l F = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z P Q R | (3) If the curl of a vector field F is zero, then F is said to be a conservative vector field. The relation between the potential function f and vector field F is F = ∇ f , where ∇ f = f x ( x , y ) i + f y ( x , y ) j . Obtain the value of c u r l F . Substitute ( 4 x 3 y 2 − 2 x y 3 ) for P , ( 2 x 4 y − 3 x 2 y 2 + 4 y 3 ) for Q and 0 for R in equation (3). c u r l F = | i j k ∂ ∂ x ∂ ∂ y ∂ ∂ z ( 4 x 3 y 2 − 2 x y 3 ) ( 2 x 4 y − 3 x 2 y 2 + 4 y 3 ) 0 | = { [ ∂ ∂ y ( 0 ) − ∂ ∂ z ( 2 x 4 y − 3 x 2 y 2 + 4 y 3 ) ] i − [ ∂ ∂ x ( 0 ) − ∂ ∂ z ( 4 x 3 y 2 − 2 x y 3 ) ] j + [ ∂ ∂ x ( 2 x 4 y − 3 x 2 y 2 + 4 y 3 ) − ∂ ∂ y ( 4 x 3 y 2 − 2 x y 3 ) ] k } = ( 0 − 0 ) i − ( 0 − 0 ) j + ( 8 x 3 y − 6 x y 2 + 0 − 8 x 3 y + 6 x y 2 ) k = ( 0 − 0 ) i − ( 0 − 0 ) j + ( 0 ) k On simplification, c u r l F = 0 Since c u r l F = 0 , F is a conservative vector and the domain of F is ℝ 2 . Thus, F is a conservative vector field. Substitute f x ( x , y ) i + f y ( x , y ) j for ∇ f in F = ∇ f . F = f x ( x , y ) i + f y ( x , y ) j (4) Compare the equation (4) and equation (1). f x ( x , y ) = 4 x 3 y 2 − 2 x y 3 (5) f y ( x , y ) = 2 x 4 y − 3 x 2 y 2 + 4 y 3 (6) Integrate equation (5) with respect to x . f ( x , y ) = ∫ ( 4 x 3 y 2 − 2 x y 3 ) d x = 4 x 4 4 y 2 − 2 x 2 2 y 3 + g ( y ) f ( x , y ) = x 4 y 2 − x 2 y 3 + g ( y ) (7) Apply partial differentiation with respect to y on both sides of equation (7)

8th Edition

ISBN: 9781305266643

Author: James Stewart

Publisher: Cengage Learning

See similar textbooks

Concept explainers

Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.

Videos

Question

Chapter 16, Problem 13RE

To determine

To show: F is a conservative vector field and evaluate the value of ∫CF⋅dr.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 16, Problem 12RE

Chapter 16, Problem 14RE

Students have asked these similar questions

Let y(t) represent your retirement account balance, in dollars, after t years. Each year the account earns 7% interest, and you deposit 8% of your annual income. Your current annual income is $34000, but it is growing at a continuous rate of 2% per year. Write the differential equation modeling this situation. dy dt

8:37 ▬▬▬▬▬▬▬▬▬ Ο Graph of f The figure shows the graph of a periodic function f in the xy-plane. What is the frequency of f? 0.5 B 2 C 3 D 8 3 of 6 ^ Oli Back Next apclassroom.collegeboard.org

2. The growth of bacteria in food products makes it necessary to time-date some products (such as milk) so that they will be sold and consumed before the bacteria count is too high. Suppose for a certain product that the number of bacteria present is given by f(t)=5000.1 Under certain storage conditions, where t is time in days after packing of the product and the value of f(t) is in millions. The solution to word problems should always be given in a complete sentence, with appropriate units, in the context of the problem. (a) If the product cannot be safely eaten after the bacteria count reaches 3000 million, how long will this take? (b) If t=0 corresponds to January 1, what date should be placed on the product?

Chapter 16 Solutions

Multivariable Calculus

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

F is a conservative vector field and evaluate the value of ∫ C F ⋅ d r .

Solution Summary: The author explains that F is a conservative vector field and evaluates the value of displaystyleint_CFcdot dr.

Concept explainers

Videos

To show: F is a conservative vector field and evaluate the value of ∫CF⋅dr.

Want to see the full answer?

Chapter 16 Solutions